Cooperative Linear Cargo Transport with Molecular Spiders

Oleg Semenov Mark J. Olah Darko StefanovicDepartment of Computer Science and Center for Biomolecular Engineering

University of New Mexico

September 15, 2012

Abstract

Molecular spiders are nanoscale walkers made with DNA enzyme legs attached to a com-mon body. They move over a surface of DNA substrates, cleaving them and leaving behindproduct DNA strands, which they are able to revisit. Simple one-dimensional models of spidermotion show significant superdiffusive motion when the leg-substrate bindings are longer-lived than the leg-product bindings. This gives the spiders potential as a faster-than-diffusiontransport mechanism. However, analysis shows that single-spider motion eventually decaysinto an ordinary diffusive motion, owing to the ever increasing size of the region of cleavedproducts. Inspired by cooperative behavior of natural molecular walkers, we propose a sym-metric exclusion process (SEP) model for multiple walkers interacting as they move over aone-dimensional lattice. We show that when walkers are sequentially released from the origin,the collective effect is to prevent the leading walkers from moving too far backwards. Hence,there is an effective outward pressure on the leading walkers that keeps them moving superdif-fusively for longer times, despite the growth of the product region. Multi-spider systems movefaster and farther than single spiders or systems with multiple simple random walkers.

1 Introduction

Molecular walkers (or, molecular motors) are natural or synthetic molecules that move over sur-faces or along tracks or fibers. Their motion is propelled by the energy released by a succession ofchemical reactions that take place as the walkers bind and release their legs; the energy is suppliedby other molecules, either from the walking surface itself or from the surrounding solution. Molec-ular walkers provide a means to transport material by non-diffusive directed motion. Molecularwalkers are ubiquitous as a transport mechanism in biological systems [14], and many of thecomplex regulatory cellular processes are controlled by the action of molecular walkers such askinesin and dynein [7]. It has been demonstrated experimentally that these natural cellular molec-ular walkers work in teams, wherein their collective action leads to behaviors not possible for asingle walker [3]. In addition, theoretical models predict that collective cooperative or competitivebehavior of walkers is fundamentally different from the behavior of individual walkers [5, 6, 8].

Inspired by the potential for walker cooperation, we propose a model describing the collec-tive behavior of teams of molecular walkers.1 Our model is based on synthetic walkers called

1A preliminary report on this research was presented at the 17th International Conference on DNA Computing andMolecular Programming [16].

Surface

Deoxyribozyme Leg (L)

Substrate DNA (S)

Product DNA (P)

Streptavidin BodySpacer

Binding

Cleavage

Figure 1: A molecular spider is a streptavidin-bodied molecular walker that moves over a sur-face of single-stranded DNA substrates. It has several flexible deoxyribozyme legs that attachto, cleave, and detach from the DNA substrates. Cleavage leaves behind shorter product DNAstrands that can still be re-bound by legs, but less strongly. (Not to scale.)

molecular spiders [12] (Sec. 2). Figure 1 shows a molecular spider walking over a surface displayingsubstrate molecules. A molecular spider has two or more enzymatic legs attached to a commonbody. The legs are deoxyribozymes—catalytic sequences of single-stranded DNA that can cleavecomplementary single-stranded DNA substrates. A spider moves over the surface, attaching to,cleaving, and detaching from the substrate sites. It leaves behind product DNA strands, which arethe surface-bound portions of the cleaved substrates. Experiments have shown that this mech-anism allows spiders to move directionally over nanoscale tracks of regularly spaced DNA sub-strates [11].

Antal and Krapivsky proposed a simple abstract model that describes molecular spider mo-tion in one dimension (1D) as a continuous-time Markov process [1, 2]. In previous work, weshowed via computer simulation and analytical arguments that walkers in the Antal-Krapivsky(AK) model move superdiffusively over significant times and distances [15], and hence are usefulas a faster-than-diffusion molecular transport mechanism. However, in the asymptotic limit oflong times, the AK walkers slow down and move as an ordinary diffusive process [1]. This can beexplained by understanding that superdiffusion is only possible while there is an energy sourceavailable to bias the motion of a spider. In the AK model, the energy is provided by the irre-versible transformation of substrates into products. This is manifested as a difference in residencetimes between substrate and product attachments. When a spider is attached to products, there isno energy available to it, and its motion is unbiased and diffusive. When a spider is attached to asubstrate, the slower rate of catalysis creates a bias that makes it more likely for the spider to moveto new, unvisited substrate sites. However, as a spider moves, it cleaves out an increasingly largeregion of products called the product sea. As the product sea grows, the spider spends increasinglymore time diffusing through it, and less time cleaving new substrate sites. Hence, the spider’smotion asymptotically decays to ordinary diffusion.

In this work, we started with the hypothesis that the collective action of many spider walkerscould lead to cooperative behavior that would enhance the superdiffusive motion of the spidersclose to the boundary. The simplicity of the AK model and the known results for single walkermotion make it a particularly useful model for extension to the study of collective spider motion.The AK model contains only those features and properties necessary to explain superdiffusivemotion of (single) spiders, abstracting away the extraneous details of the specific spider chemistry

2

and focusing on the kinetics of nearest-neighbor walking gaits for multipedal enzymatic walkers.We extend these nearest-neighbor walking gaits to collective motion of spiders in 1D. An immedi-ate consequence is that AK spiders cannot move past one another, as their legs are too short to hopover adjacent walkers. As a result even if the product sea becomes very large, a spider near theboundary of the sea cannot move further back than the next spider behind it. Hence by adding spi-ders at the origin the effective size of the product sea as seen by the leading spiders is reduced. Inessence, the presence of other spiders can act as an exclusionary pressure that enhances transportof cargo carried by the leading spiders. Ideally the leading spider would perceive a constant-sizesea of products behind it, and its diffusive excursions into that sea would be of constant duration,which would lead to asymptotically superdiffusive motion.

In Sec. 3 we present a model for cooperative multi-spider transport over an infinite 1D lattice.The model relies on an unlimited supply of identical new spiders injected at the origin when-ever the sites at the origin are unoccupied. Conceptually this spider injection approximates therelease of spiders from a large reservoir near the origin. Using Kinetic Monte Carlo (KMC) meth-ods, we show that multi-spider systems exhibit significantly superdiffusive motion within thetime bounds studied (Sec. 4), with both the duration of the superdiffusive period and the peaksuperdiffusivity coefficient greater than for single walkers (Sec. 4.2). This shows potential formulti-spider systems with injection to be used to perform useful tasks in nanoscale computationaland communication systems by providing a faster-than-diffusion mechanism of “first-to-target”transport.

However, even with an unbounded reservoir of new spiders, the asymptotic behavior of theleading spider is still ordinary-diffusive, not superdiffusive! Sections 5–7 explain this result byinvestigating the effective size of the product sea, the number of injected spiders, and the spa-tial distribution of spiders. Importantly, we find that even though the multi-spider model is notasymptotically superdiffusive, it still has superior transport properties. Indeed, the multipedalnature of spiders has quantifiable advantages over systems of interacting simple random walkers(i.e., spiders with just one leg). In analogy to the bipedal walkers, ubiquitous in natural systems,multipedal walkers present distinct advantages for molecular transport, whether acting alone oras part of a collective.

2 Molecular Spiders

A molecular spider (Fig. 1) has a rigid, chemically inert body (such as streptavidin) and severalflexible legs made of deoxyribozymes—enzymatic single-stranded DNA that can bind to andcleave complementary strands of a DNA substrate at the point of a designed ribonucleic baseimpurity. When a spider is placed on a surface on which the appropriate DNA substrate has beendeposited (or nanoassembled), the legs bind to the substrate and catalyze its cleavage, creatingtwo product strands. The upper portion floats away in solution and we do not consider it further.The lower portion remains on the surface, and, because it is complementary to the lower partof the leg, there is some residual binding of the leg to the product, typically much weaker andshorter-lived than the leg-substrate binding. The leg kinetics are described by the five reactionsin Eq. 1 relating legs (L), substrates (S), and products (P), in which we have folded the catalysis

3

reaction and subsequent dissociation reactions into a single kcat rate:

L + Sk+S−→←−k−S

LS kcat−→ L + P (1)

L + Pk+P−→←−k−P

LP (2)

2.1 The Antal-Krapivsky Model

The Antal-Krapivsky model [1, 2] is a high-level abstraction that represents a molecular spideras a random walker. Unlike ordinary random walkers, each AK spider has k legs. The chemicalactivity of the multiple legs is independent, but their motion is constrained by their attachment toa common body; in the model, any two legs must be within distance s. The legs walk over siteson a regular 1D lattice, where each site is either a substrate or a product.

Mathematically, the AK model takes the form of a continuous-time Markov process. The sys-tem states of this process are given by the combined state of the lattice sites and the state of thespider legs. All lattice sites are initially substrates and are only transformed to products when aleg detaches from the substrate (via catalysis). Thus the state of the lattice sites is fully describedby the set P ⊂ Z of product sites. The state of the spider is completely defined by the set F ofattached feet locations. Thus any state can be described as the pair (P, F).

2.1.1 Spider Gait

We call F a configuration of the legs. The gait of a spider is defined by what configurations andwhat transitions between configurations are allowed in the model. In any state (P, F) ∈ Ω whereΩ is a state space of the Markov process, all k legs are attached. Together with the restriction thatat most one leg may be attached to a site, this implies that

|F| = k. (3)

Additionally, the legs are constrained by their attachment to a common body. If the spider hasa point body with flexible, string-like legs of length s/2, then any two feet can be separated by atmost distance s, thus,

max(F)− min(F) ≤ s. (4)

The transitions in the spider Markov process correspond to individual legs unbinding andrebinding. When a spider is in configuration F, any foot i ∈ F can unbind and move to a nearest-neighbor site j ∈ i + 1, i − 1 to form a new configuration F′ = (F \ i) ∪ j, provided the newconfiguration does not violate one of the constraints of Eqs. 3 and 4. A transition i → j is calledfeasible if it meets these constraints. The feasible transitions determine the gait of the spider. Thenearest-neighbor hopping combined with the mutual exclusion of legs leads to a shuffling gait,wherein legs can slide left or right if there is a free site, but legs can never move over each other,and a leg with both neighboring sites occupied cannot move at all. If the legs of such a spiderwere distinguishable, they would always remain in the same left-to-right ordering.

In the case where k = 2, s = 2 the gait of the spider is particularly simple (Fig. 2), but it stillexhibits constraints on the coordination of the leg motion that are not present for k = 1 walkers.

4

︸ ︷︷ ︸

s = 2

k = 2

0 1 2 3 4

0 1 2 3 4 0 1 2 3 4

rate=1

SubstrateProduct

rate=r

cleavage

Figure 2: An AK spider with two legs (k = 2) and maximum leg spacing s = 2 is attached to aproduct at site 1 and a substrate at site 3. It can detach from the product at rate 1, after which itcan only move to site 2 without violating Eq. 4. Alternatively it can cleave and detach from site 3at rate r, after which it must move to site 2.

2.1.2 Transition Rates

The rate at which feasible transitions take place depends on the state of the site i. If i is a product,the transition rate is 1, but if i is a substrate, the transition occurs at a slower rate r < 1. Thisis meant to model the realistically slower dissociation rates from substrates, corresponding tochemical kinetics where kcat/k−P = r < 1. The effect of substrate cleavage is also captured in thetransition rules. If for state (P, F), where i ∈ F \ P, the process makes the feasible transition i → j,then the leg will cleave site i before leaving, and the new state will have P′ = P ∪ i.

The relation of the AK model to the chemistry of the spiders in Eq. 1 can be understood if oneassumes the chemical rates are given as in Eq. 5:

k+S = k+P = ∞k−S = 0k−P = 1

kcat = r < 1

(5)

The infinitely fast on-rates account for all legs always being attached; when a leg unbinds it in-stantly rebinds to some neighboring site. Thus the spider is modeled as jumping from configura-tion F to configuration F′.

2.2 Superdiffusive Motion of Single AK Spiders

To characterize the motion of spiders we use the notion of superdiffusion. Superdiffusive motioncan be quantified by analyzing the mean squared displacement (MSD) of a spider as a functionof time. For diffusion in a one-dimensional space with diffusion constant D, the mean squareddisplacement is given by Eq. 6.

5

msd(t) =⟨

x2(t)⟩= 2Dtα

α = 0 stationary0 < α < 1 subdiffusiveα = 1 diffusive1 < α < 2 superdiffusiveα = 2 ballistic or linear

(6)

We say that the spider is moving instantaneously superdiffusively [10] at a given time t if

α(t) =d(ln (msd(t)))

d(ln (t))> 1. (7)

Using Kinetic Monte Carlo simulations [4] of the Markov process we can estimate the msd(t)for the spider process for different parameter values by averaging over many realizations x(t) ofthe process X(t), where each x(t) is a function from t ∈ [0, tmax] to the state space Ω of the spiderprocess, and x(t) ∼ X(t).

When the rate of substrate cleavage is slower than the rate of product detachment (r < 1), eachspider process goes through three different phases of motion defined by their instantaneous valuefor the exponent α of msd(t). Initially spiders are at the origin and must wait for both legs to cleavea substrate before they start moving at all, so when t < 1/r the process is essentially stationary;we call this largely unimportant phase the initial phase. After the spiders take several steps, spiderswith r < 1 show a sustained period of superdiffusive motion over many decades in time. We callthis the superdiffusive phase, and define it as the period during which the instantaneous estimateof α > 1.1. The cutoff of α = 1.1 is somewhat arbitrary but represents a threshold where spidersare moving significantly superdiffusively, in contrast to spiders with r = 1, which never haveα > 1. Finally, Antal and Krapivsky showed that as time goes to infinity, all spiders will approachordinary diffusion with α ≈ 1. This final stage is called the diffusive phase and is characterized byspiders mainly moving over regions of previously cleaved products, which makes the values of rirrelevant, since all spiders move with rate 1 over product sites.

2.2.1 The Boundary and Diffusive Metastates

To explain this behavior we observe that spiders with s = 2 and k = 2 always cleave all sitesthey move over since the AK model does not permit legs to change their effective ordering onthe surface. Hence, the spiders move with a shuffling gait, consuming all products in the regionthey move over. This leads to the formation of a sharp boundary between a contiguous region ofproducts called the product sea, and the remainder of the unvisited sites which are still substrates.The product sea has left and right boundaries defined as

bR = max (P) + 1, and, bL = min (P)− 1. (8)

Hence the sites bR and bL are substrates, and moreover, they are the only substrates a spider withk = 2, s = 2 can reach.

Given this definition of boundary, we can consider the spider Markov process as moving be-tween two metastates. If the state of the system is X = (P, F), then if bR ∈ F or bL ∈ F, we saythe system is in the boundary metastate (X ∈ B), otherwise the system is in the diffusive metastate

6

(X ∈ D). When the system is in the B metastate, the spider moves ballistically towards unvisitedsites; when it is in the D state, the spider’s motion is ordinary diffusive. We define a B-period asan interval of time during which the spider is in the B state and a D-period as an interval of timeduring which the spider is in the D state. A realization of the spider process alternates betweenB- and D-periods. We define the distribution of the durations of B-periods that begin at time t asτB(t), and the distribution of durations of D-periods beginning at t as τD(t)

The transitions from the B state to D state are independent of the previous state of the systembefore it entered the B state, hence 〈τB(t)〉 is independent of time. However, the transitions fromthe D state back to the B state depend on the size of the product sea that the spider has left behind,and this size increases with time. Correspondingly, it has been shown that 〈τD(t)〉 = Θ

(√t)

[15].This explains the transient superdiffusion at short times when the spider spends more time in

the B state, and the decay to ordinary diffusion at long times, as the spider spends more and moreof its time in the D state.

There are two options to increase the superdiffusive effect of the spider motion: (1) decreasethe effective size of the product sea, and hence the time needed to escape from it and return tothe boundary (〈τD(t)〉); or (2) decrease the rate at which spiders leave the boundary. Here wefocus on option (1), by means of localized release of spiders at the origin, which effectively fills theproduct sea with follower spiders, preventing the leading spiders from moving as far backwardsaway from the boundary. This works because spider legs cannot occupy the same site at thesame time, and spiders walk with a shuffling gait, sliding left or right one site at a time, so aspider cannot jump over an adjacent spider. The presence of multiple spiders will restrict themotion of the spiders around them, reducing the effective size of the product sea as seen by awalker at the boundary. By releasing many spiders sequentially at the origin we make the D stateof the leading spiders shorter. Ideally, we might desire to make the duration of each diffusiveexcursion of the leading spider independent of the number of sites that have already been cleaved(i.e. 〈τD(t)〉 = O(1)). Such a system would have the potential for asymptotically superdiffusivemotion, but it will turn out that this is not achieved in the multi-spider injection model.

3 The Multi-Spider Model

Cooperative and interactive behavior of spiders can be studied by extending the AK model to amulti-spider model that describes a system of S ≥ 2 spiders moving simultaneously. Every spiderin the multi-spider model has identical values for parameters k (the number of legs) and s (themaximum leg spacing), and they all move over the same 1D surface of substrates. The (otherwiseindistinguishable) spiders are enumerated as 1, . . . , S, which allows the state of the system to bedescribed as

X = (P, F1, ..., FS). (9)

Here, Fi ⊂ Z gives the attached leg locations of spider i. In analogy to Eqs. 3 and 4, we maintainfor all i that

|Fi| = k (10)

andmax(Fi)− min(Fi) ≤ s. (11)

To extend the chemical exclusionary properties of spider legs to multi-spider systems, we addthe restriction that any site on the surface can be occupied by only one leg of any spider, so that

7

for all i, j:Fi ∩ Fj = ∅. (12)

Finally, we define a mechanism to allow the addition of new spiders into the system, allowingS to grow with time, while maintaining the Markovian properties of the process. New spiders canbe added at an injection site I = 0, 1 for any state in which the sites 0 and 1 are unoccupied. Anew spider is added as a Poisson process with rate λ > 0, and the initial state of the new spideris FS+1 = 0, 1. In the limit when λ = ∞, a new spider is added as soon as the injection siteis unoccupied. Even in this limiting case, the presence of other spiders at the injection site andtheir finite rate of movement out of this site constrain the multi-spider system to a finite numberof spiders at all times.

3.1 Rebinding Gait.

With multiple spiders on a lattice, there are situations where a particular spider is completelyblocked from movement. This happens when its legs are together (i.e., on adjacent sites) andother spiders occupy the sites to its immediate left and right. Thus, to simplify the Markov processdescription, we introduce a slight change to the gait of the walker. When a leg detaches from a sitei it can move not only to sites i − 1 and i + 1 as in the AK model, but also back to site i. It choosesfrom any site in i − 1, i, i + 1 with equal probability, provided none of the new configurationsviolates the constraints of Eqs. 10, 11, and 12. Thus, even if sites i − 1 and i + 1 are occupied,the leg has somewhere to go. We call this new spider gait the rebinding gait. It ensures that theprobability that any particular leg will move is independent of the state of the rest of legs in thesystem, which simplifies the Monte Carlo simulation of the system.

Furthermore, the rebinding gait is more chemically realistic, as the enzymatic leg of a realmolecular spider can always rebind to the site it just dissociated from, and its detachment shouldbe independent of the state of the rest of the system.

From an analytical perspective, the effect of allowing rebinding is to slow the movement ofwalkers because of the potential for dissociations that do not move a spider leg. However, simu-lation results extended to very long times (Fig. 3) show that this change in effective rates does notqualitatively change the motion of a single spider. The rebinding gait leads to a constant-factordecrease in the mean squared displacement. In the remainder of the paper when comparisons ofmulti-spider systems with rebinding gait are made with single-spider systems, the single-spidersystems have the normal AK gait, without rebinding. This gives them a constant factor advan-tage; however, as will be seen, even with this handicap the multi-spider systems are superior astransport devices.

4 Simulation Results for the Multi-Spider Model

The multi-spider system provides a simple model for cooperative transport using interactingwalkers. In this application, walkers start at the origin of a 1D surface covered with energy-supplying substrate. The walkers move outwards in the plus and minus directions consumingsubstrate to bias their motion outwards, leaving an ever increasing sea of products (P ⊂ Z) inbetween the farthest sites visited in the plus and minus directions. Because spiders always cleavesubstrates when they detach from a site, and the spiders with s = 2, k = 2 have a shuffling gait

8

100 101 102 103 104 105 106 107 108

Time

10-1

100

101

102

103

104

105

106

107

108

msd

(t)

r = 0.1

r = 0.1 rebind

time squared

diffusion

Figure 3: Estimate of msd(t) for spiders with the AK gait and the rebinding gait with k = 2, s = 2,and r = 0.1.

wherein they cannot hop over any substrates, the sea of products P is always contiguous and in-cludes the origin. Thus, as explained in Sec. 2.2.1, there is a well defined concept of a left (bL) andright (bR) boundary between the product sea and the unvisited substrates.

In the multi-spider model, the spiders fill this product sea, creating an exclusionary pressurethat prevents the outermost spiders from moving past them. At any given time there will be oneleftmost 1 ≤ Ls ≤ S and one rightmost spider 1 ≤ Rs ≤ S. We are interested in the location ofthese leading Ls and Rs spiders as the system evolves. When the Markov process is in state Xas given by Eq. 9, the i-th spider’s position is defined as the mean of Fi. We use a function µ todescribe the position of any spider 1 ≤ i ≤ S as

µ(i) = ∑ Fi

k. (13)

Note that when k = 2, s = 2 as in the multi-spider model, µ(i) only takes on half-integer valuesand the value of µ(i) uniquely determines the value of Fi. This is in general not the case for largervalues of k and s.

It is sometimes possible for the identities of the leftmost or rightmost spider to change. Whenµ(Ls) > 2, all the spiders are to the right of the injection point I = 0, 1, and when µ(Rs) < 0.5,all the spiders are to the left of I. In these cases, when a new spider is injected at I, it becomes thenew Ls or Rs, respectively. This situation becomes very unlikely as the number of spiders releasedincreases; however, when reporting the MSD of Ls and Rs, these identity changes are importantand are accounted for in our analysis.

4.1 Experimental Setup

To quantify the transport characteristics of the multi-spider injection system, we define a multi-spider system that begins with two spiders on either side of an injection site near the origin. The

9

︸ ︷︷ ︸

Rs

0 1 2 3 4-1-2-3-4

Ls

I = 0,1

Product Sea (P)

SubstratesSubstrates

bL bR

λ=∞

Figure 4: The initial configuration used for the multi-spider model simulations.

Table 1: Parameters studied for the multi-spider model

Parameter Description

k = 2 number of legss = 2 maximum leg separationP = −2,−1, 0, 1, 2 initial product seaS = 2 initial number of spidersF1 = 1, 3 initial location of rightmost spiderF2 = −3,−1 initial location of leftmost spiderI = 0, 1 injection siteλ = ∞ injection ratek−P = 1 rate of detachment from productskcat = r ≤ 1 rate of detachment from substrates

two initial spiders start on the boundary on either side of an initial product sea (Fig. 4). The preciseparameters studied are given in Table 1.

We use the Kinetic Monte Carlo (KMC) method [4] to numerically sample traces of the multi-spider Markov process. In Table 2 we show the number of runs of the Markov process sampledfor each r-value and the minimum time each sample was run until.

The primary parameter of interest is the chemical kinetics parameter r. When r = 1 there is noeffective chemical difference between substrates and products, and hence no energy is availablein the substrates to bias the outermost spiders’ motion. When r < 1 the chemical difference at theboundary acts to bias the motion of the leftmost spider (Ls) and the rightmost spider (Rs) awayfrom the origin when they are at their respective boundaries. This creates effective superdiffusionfor the leading spiders as long as they stay near the boundary.

Table 2 shows that we have many fewer simulation traces for tmax = 108 than for tmax =107. Indeed, these simulation counts are far fewer than the 5000 traces computed for the singlespider model [15]. We were not able to compute more traces because simulation of the multi-spider model requires much more computational resources for large values of tmax near 108. Our

10

Table 2: Number of KMC runs for each parameter value and minimum simulation time combina-tion.

r-valuetmax 1.0 0.5 0.1 0.05 0.01 0.005

1.0 × 107 1800 1800 1800 1800 1800 18001.0 × 108 200 - 200 - - -

KMC simulations of the spider systems consist of iterative computation of consecutive discreteevents of the underlying Markov process. Every individual event takes a constant amount ofcomputational (wall) time, but the simulation time intervals between events are exponentiallydistributed based on the total rate of all potential transitions from the current state. There aretwo possible transitions from every state of the single spider model: the left leg moves, or theright leg moves. Thus, the mean simulation time duration between events remains within 1/2and 1/(1 + r) for every simulated step. Since r is a constant that does not depend on tmax, theexecution time of our single spider KMC algorithm is O(tmax), i.e., it depends linearly on thesimulation time. However, in the multiple spider model the mean duration between events is notconstant. When new spiders are injected, the number of different possible events in the systemincreases, and so the simulation time intervals between those events become smaller. Thus, toachieve desired maximum simulation time tmax for the multi-spiders model, we need to simulatemore discrete events than for the single spider model. In section 5.2 we estimate that the number ofreleased spiders grows as O

(√t)

. Hence, the execution time of the multi-spider KMC simulation

algorithm is O(t3/2max

).

4.1.1 Observed Superdiffusion of the Leading Spiders.

As discussed in Sec. 2.2, single spider systems with r < 1 show transient superdiffusive behav-ior [15]. Single spiders move faster than ordinary diffusion for a significant time and distance, buteventually slow down and move as an ordinary diffusion. Hence, the leading spiders (Ls and Rs)of the multi-spider model also should move superdiffusively when they are near the boundary.This can be quantified by estimating the mean squared displacement of the leading spiders inthe multi-spider model. Because the environment and the walker are symmetrical, we can, with-out loss of generality, represent the mean squared displacement of the outermost walkers by theposition of the rightmost spider,

msd(t) = 〈µ(Rs)(t)〉. (14)

Figure 5 shows the KMC simulation estimate of msd(t) for the multi-spider model on a log-logplot for each measured r parameter value. In this plot, straight lines correspond to power laws,that is, to Eq. 6, and the parameter α is given by the slope. To show the instantaneous value of α,we use finite difference methods to estimate α(t) as in Eq. 7. Figure 6 shows the result of using theSavitzky-Golay smoothing filter [13] on these estimates of α(t).

These results show the same three-phase behavior as observed in single spider simulations(Sec. 2.2). The phases can be observed by noting the estimate of the value of α(t) in relation to thehorizontal line in Fig. 6 representing α(t) = 1.0. Below this line, the leading spiders are moving

11

100 101 102 103 104 105 106 107

Time

10-1

100

101

102

103

104

105

106

107

108

msd

(t)

r=1.0r=0.5r=0.1r=0.05r=0.01r=0.005time squareddiffusion

Figure 5: Mean squared displacement for Rs. Reference lines are shown for ordinary diffusion andballistic motion.

100 101 102 103 104 105 106 107

Time

0.2

0.4

0.6

0.8

1.01.11.2

1.4

1.6

1.8

2.0

(t)

r=1.0r=0.5r=0.1r=0.05

r=0.01r=0.005td

max value

Figure 6: Smoothed finite difference approximation of α(t) for Rs. Horizontal lines define thethreshold for ordinary diffusion at α = 1.0 and our defined threshold for superdiffusion at α = 1.1.

12

Table 3: Properties of the MSD and the superdiffusive regime defined by α(t) > 1.1 for Rs. Resultsare compared for the multi-spider model as well as the single spider model (Sec. 4.2) and the k = 1spiders which are simple random walkers (Sec. 6).

Multi-spider r-value Single-spider r value k = 1 spider r value

1.0 0.5 0.1 0.05 0.01 0.005 0.05 1.0

αmax 1.38 1.37 1.63 1.73 1.87 1.93 1.43 1.41tαmax 9.03 × 101 1.03 × 102 8.60 × 101 1.50 × 102 2.83 × 103 1.02 × 104 2.17 × 102 2.71 × 101

td 2.43 × 105 2.75 × 105 2.10 × 105 1.89 × 105 1.02 × 106 3.31 × 106 2.52 × 104 1.14 × 105

subdiffusivly and above this line the leading spiders are moving superdiffusivly. For each valueof r, the systems exhibit the following three sequential phases:

1. The initial phase is defined when t < 1/r. At these times, very few steps have been made,and the value of α(t) is largely dependent on peculiarities of the initial configuration, andnot of relevance to transport phenomena.

2. The superdiffusive phase begins when t > 1/r and continues while the α(t) ≥ 1.1. Duringthis phase the spiders move significantly faster than diffusion. Decreasing values of r leadto increasing maximum values of α(t), and a longer time until the spider returns to theα(t) = 1.1 threshold.

3. The diffusive phase begins when the instantaneous value of α(t) = 1.1, and continues indef-initely, as even the leading spiders eventually spend almost all of their time diffusing overthe products instead of cleaving new sites at the boundary.

Thus, even though a multi-spider system adds spiders at the origin as fast as possible to pre-vent the leading spiders from moving too far backwards into the product sea, all multi-spidersystems eventually decay to diffusion. This is the same qualitative behavior exhibited by singlespider systems. However, multi-spider systems move superdiffusively over significantly longertimes, and even when r = 1.0. They also reach a higher peak value of α(t). The superdiffusiveproperties of the multi-spider model can be quantified by examining the following:

• αmax = maxt≥0 (α(t)), the peak instantaneous value of α(t), which should satisfy 1 ≤ αmax ≤2;

• tαmax = argmaxt≥0 (α(t)), the time at which the peak of α(t) is reached;

• and td, the time at which α(t) drops below the threshold of 1.1, and enters the diffusivephase.

The estimates of these quantities are given in Table 3. The results show that αmax and td gener-ally increase with decreasing r. Furthermore, the walkers with r = 0.005 have peak α(t) valuesabove 1.9 and remain superdiffusive over times more than 6 orders of magnitude larger than themean leg-product residence time.2 Hence, for finite distances and times relevant to most transportprocesses the multi-spider model can achieve significant superdiffusive motion, and with small rvalues becomes nearly ballistic for significant spans of time and distance.

2The value of k−P is a free parameter in the model. We choose time units so that rate k−P is normalized to 1, hence alltime units are measured relative to 1/k−P = 1.

13

100 101 102 103 104 105 106 107

Time

10-3

10-2

10-1

100

101

msd

(t)/2t

r=1.0r=0.5r=0.1r=0.05r=0.01r=0.005

Figure 7: Effective diffusion rate (D(t)) for Rs.

A further method of characterizing the transport behavior of multi-spider systems is to look attheir effective instantaneous diffusion rate which (in 1D) is defined as

D(t) = msd(t)/2t. (15)

This is arrived at by setting α = 1 in Eq. 6. The value of D(t) can be thought of as the diffusion ratea simple diffusing particle would need to have the same mean squared displacement as the spidersystem at time t. Hence, greater values of D(t) correspond to faster transport systems. Figure 7shows D(t) for the multi-spider systems. While initially the spiders with lower r have larger Dvalues, eventually the spiders with the smallest r values are superior.

4.2 Comparison with Single Spiders

The multi-spider systems can be directly compared with the single spiders to understand exactlyhow useful the additional interior spiders are for transport. Figure 8 shows the results of com-paring the msd(t) for a single spider, and the leading spider of the multi-spider model, both withr = 0.05. There is a significant transport advantage for the multi-spider system. Furthermore,Fig. 9 shows the estimate of α(t) for these two systems, and the leading spider of the multi-spidersystem maintains a higher value of α(t) at all times, and significantly longer superdiffusive period.The values of αmax and td are summarized in Table 3, and the multi-spider system is superior inboth measurements. Finally, we compare the first passage times for the single versus multi-spidermodels in Fig. 10. The mean first passage time 〈fpt(d)〉 is the average time for the leading spiderto first visit a site at a distance d from the origin. At large times the multi-spider systems have alarge advantage in this key transport statistic. Overall, the comparison with the single AK spidershows that for distances less than 3000 sites from the origin the leading spider of the multi-spidersmodel reaches unvisited sites up to 5.25 times faster than a single AK spider.

14

100 101 102 103 104 105 106 107

Time

10-1

100

101

102

103

104

105

106

107

108

msd

(t)

leading spider of multiple spiders, r=0.05single spider, r=0.05

Figure 8: Mean squared displacement for the leading spider Rs in the multi-spider model versusa single AK spider.

100 101 102 103 104 105 106 107

Time

0.6

0.8

1.0

1.1

1.2

1.4

1.6

1.8

(t)

leading spider of multiple spiders, r=0.05single spider, r=0.05max valuetd

Figure 9: Finite difference approximation of α(t) for the leading spider Rs in the multi-spidermodel versus a single AK spider.

15

0 500 1000 1500 2000 2500 3000Max absolute dist

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Mean T

ime

×107

leading spider of multiple spiders, r=0.05single spider, r=0.05

Figure 10: Comparison of the first passage time of the leading spider Rs in the multi-spider modelversus a single AK spider.

5 Asymptotic Diffusion in the Multi-spider Model

In comparison with a single spider, the simulation results in Sec. 4 show that multi-spider systemsexhibit larger values for D, αmax, and td—all essential measures of transport efficiency. However,multi-spider systems still eventually decay to diffusion despite the unlimited supply of spidersinjected at the origin. By analogy with the single spider model (Sec. 2.2), this implies that theeffective size of the product sea as seen by the leading spiders is not constant. As the effectiveproduct sea grows, the leading spiders spend progressively less time at the boundary in the B statewhere they move ballistically, and more time in the diffusive D state where they move over theproduct sea. This leads to a value of α(t) → 1.0, as t → ∞, and the leading walkers are effectivelydiffusing. The origins of this effect can be understood in detail by examining the effective size ofthe product sea (Sec. 5.1), the number of released spiders (Sec. 5.2), and their spatial distribution(Sec. 5.3).

5.1 Effective size of the product sea

In 1D, spiders cannot move past each other, and thus the motion of the leading spiders Rs andLs is constrained by the presence of their neighboring spiders. In particular when S ≥ 3 we candefine the next-leading spiders, R′

s and L′s as

R′s = max

1≤i≤Si 6=Rs

(µ(Fi)) (16)

andL′

s = min1≤i≤Si 6=Ls

(µ(Fi)) . (17)

16

︸ ︷︷ ︸

Neff

RsR′

s

bRProduct Sea Substrates

AbsorbingBoundary

ReflectingBoundary

Figure 11: The effective size of the product sea (Neff) as seen by leading spider Rs is the areabetween next-leading spider R′

s and the right boundary bR.

When S ≥ 4, L′s 6= R′

s, and the remaining S − 4 spiders are called the interior spiders. The impor-tance of the next-leading spiders, R′

s and L′s, is that they define the effective size of the product sea

as seen by the leading spiders. Without loss of generality we focus only on the rightmost spidersRs and R′

s and define〈Neff(t)〉 = 〈max

(FR′

s(t)

)+ 1 − bR(t)〉, (18)

where the system state at time t is given by X(t) from Eq. 9, and bR(t) is the right boundaryfor that state as defined in Eq. 8. Figure 11 illustrates what Neff is for a particular state X. Thechoice of name for 〈Neff(t)〉 is meant to correspond to Antal and Krapivsky’s 〈N(t)〉 [1], whichis the mean number of sites cleaved by a single spider system at time t. It has been shown forsingle spiders that 〈N(t)〉 = Θ

(√t)

[1], and that this implies that the time to leave the diffusive

metatstate 〈τD(t)〉 = Θ(√

t)

[15]. Since 〈τD(t)〉 grows with time, and 〈τB(t)〉 does not, singlespiders eventually spend almost all their time diffusing in the D state and hence their motion isasymptotically diffusive.

The relation of 〈N(t)〉 with 〈τD(t)〉 is based on the mathematics of the mean time for a randomwalker to escape an interval of size N with two absorbing boundaries. This gives the time for awalker to leave the product sea and return to the boundary. In the case of the multi-spider system,Neff is also meant to represent the size of the region of products to escape from; however, it has onereflecting boundary at max

(FR′

s

)and one absorbing boundary at bR. The problem of escape from

a region with one reflecting and one absorbing boundary is equivalent to escape from a region oftwice the size with two absorbing boundaries (Fig. 12). Hence, we note the relation:

〈Neff(t)〉 = 〈N(t)〉/2. (19)

Further analysis of this relationship is deferred to Sec. 7. However, the importance of Neff canbe understood simply—if 〈Neff(t)〉 increases with time, then because 〈τD(t)〉 = Θ(〈Neff(t)〉), theleading spider Rs must eventually move diffusively. This is indeed the case for all values of r, asshown in Fig. 13. In this figure we used the Levenberg-Marquardt algorithm [13] to fit power lawsto the estimates of 〈Neff(t)〉. We see that, interestingly, the exponents are close to 0.5—exactly aswith single spiders.

Thus, 〈τD(t)〉 for Rs also grows with time and therefore leads to asymptotic diffusion. How-ever, in the multiple spider model the effective size of the product sea is much smaller than thetotal number of products (〈Neff(t)〉 ¿ |P(t)|) because most of the product sea is filled with otherspiders, whereas in the single spider model 〈N(t)〉 = |P(t)|. Thus, Rs in the multi-spider modelremains superdiffusive for much higher values of |P| than a single spider does. Fig. 14 comparesthe effective size of the product sea of the multi-spider model with the number of products of the

17

︸ ︷︷ ︸

Neff

Rs

Neff

︸ ︷︷ ︸

N = 2Neff

︸ ︷︷ ︸

ReflectingBoundary

AbsorbingBoundary

AbsorbingBoundary

AbsorbingBoundary

AbsorbingBoundary Reflection

Product Sea

Product Sea

Effective product sea in the single spider model

Effective product sea in the multi-spider model

bR

bL

Substrates

Substrates

bR

Figure 12: The problem of escape from an area of size Neff with one reflecting and one absorbingboundary is equivalent to the problem of escape of a single spider from a region of size N = 2Neffwith two absorbing boundaries.

0.0 0.2 0.4 0.6 0.8 1.0Time ×107

0

200

400

600

800

1000

1200

Neff

r=1.0, data

r=1.0, Neff =0.37t0.46

r=0.5, data

r=0.5, Neff =0.46t0.47

r=0.1, data

r=0.1, Neff =0.46t0.48

r=0.05, data

r=0.05, Neff =0.57t0.47

r=0.01, data

r=0.01, Neff =0.29t0.51

r=0.005, data

r=0.005, Neff =0.14t0.55

Figure 13: The effective size of the product sea 〈Neff(t)〉 grows with time, and hence leads toasymptotically diffusive motion for the multi-spider model.

18

0.0 0.2 0.4 0.6 0.8 1.0Time ×107

0

500

1000

1500

2000

2500

Neff

leading spider of multiple spiders,r=0.05single spider, r=0.05

Figure 14: The effective size of the product sea (〈Neff(t)〉) as seen by Rs in the multi-spider modelversus the effective size of the product sea (〈N(t)〉) for a single walker in the AK model.

AK model. While both 〈N(t)〉 and 〈Neff(t)〉 grow with time, the single spider sees a much largerproduct sea, which explains the results of Sec. 4.2.

5.2 Number of Released Spiders

With injection rate λ = ∞, spiders in the multi-spider model are released at I = 0, 1 wheneverpossible, yet the presence of other spiders in the sites 0 and 1 prevents release, and so keeps thetotal number of spiders, S, finite. Hence, S(t) is a random variable giving the number of spidersreleased by time t. Figure 15 shows our estimates for 〈S(t)〉, for each studied value of r. Again weused the Levenberg-Marquardt algorithm to fit power laws, and observe approximate exponentsof 0.5. Interestingly, 〈S(t)〉 appears to be dependent on r only initially, whereas at later times thevalues of 〈S(t)〉 for all r are nearly identical. Thus, the release of spiders, which occurs at theorigin and away from the substrates at the boundaries, is independent of r, which limits the totalnumber of released spiders regardless of how fast the leading spiders move.

5.3 Spatial Distribution of Spiders

Until now we have focused mainly on the position of the leading spiders Rs and Ls, but the be-havior of the interior spiders controls the release of spiders at the origin and the effective productsea size near the boundary. At any time t, we measure the density of spiders using a histogramwith 100 equally spaced bins over the maximum positions obtained by any spider in any simula-tion trace at that time. Each bin with n sites can contain at most n/2 spiders, hence the maximumdensity for each bin is 0.5. Figure 16 shows the density of spiders at tmax = 107 for r = 1 andr = 0.05. The density of spiders near the origin is approximately the same in both cases and isnearly equal to the 0.5 maximum possible density. This explains why 〈S(t)〉 for large t is nearlythe same for all r values. The sites near the origin are very densely packed for any r-value and

19

100 101 102 103 104 105 106 107

Time

100

101

102

103

104

S(t

)

r=1.0, data

r=1.0, S =0.54t0.51

r=0.5, data

r=0.5, S =0.54t0.51

r=0.1, data

r=0.1, S =0.54t0.51

r=0.05, data

r=0.05, S =0.54t0.51

r=0.01, data

r=0.01, S =0.51t0.52

r=0.005, data

r=0.005, S =0.45t0.53

Figure 15: Mean number of released spiders, 〈S(t)〉; the log-log scale is necessary to exhibit thedifferences at early times.

10000 5000 0 5000 10000Position

0.0

0.1

0.2

0.3

0.4

0.5

Density

(a)

10000 5000 0 5000 10000Position

0.0

0.1

0.2

0.3

0.4

0.5

Density

(b)

Figure 16: Mean spider density at tmax, for r = 1 (a), and r = 0.05 (b).

20

(a) (b)

Figure 17: Evolution of mean spider density through time, for r = 1 (a) and for r = 0.05 (b).

the passive addition of spiders is not presented with many opportunities to add new spiders evenwhen the injection rate is infinite.

The density of spiders falls off nearly linearly away from the origin, until approximately dis-tance 4000 (at tmax = 107), where densities for both r values gradually transition to long tails withnearly 0 density. Essentially, the only difference between the densities for r = 1 and r < 1 is atthe tails. The tails for r = 0.05 are much longer, corresponding to the greater msd(tmax) valueobserved for this r-value. This is to be expected as only the leading spiders Rs and Ls ever see asubstrate, and all other spiders only move over products. Thus the rate r only affects the motionof the leading spiders and those spiders near to the leading spiders that have comparatively morespace to diffuse in. The vast majority of the interior spiders are too far from the leading spidersto see the effects of the r-value. Hence, the distribution of spiders away from the boundaries isnearly identical for r = 1 and r < 1.

This similarity in densities of interior spiders is present at all times, as can be seen in Fig. 17,which shows the evolution of these densities with time. For both r = 1 and r = 0.05, the densitiesdrop off linearly around the origin, until a critical point where they transition gradually to thenear-zero density tails. The tails of the r = 0.05 spider density remain longer as expected basedon the MSD results of Fig. 5, but the evolution of the interior spider density is nearly independentof r.

5.3.1 Distribution of Spider Strides

Spiders with k = 2, s = 2 only have two possible foot patterns: either both of their legs aretogether, or both are apart (Sec. 2.1). The together and apart patterns can be called the two possiblestrides of a spider. A single spider is equally likely to be in either of the two strides. However, thespiders in the multi-spider model show a curious distribution of leg patterns. Figure 18 shows thedistribution of spiders in each of the two strides at time tmax, and again the results are remarkablysimilar for both r = 1 and r < 1. The high density of spiders near the origin leads to a very strongbias towards the together stride.

21

10000 5000 0 5000 10000Position

0.0

0.1

0.2

0.3

0.4

0.5

Density

(a)

10000 5000 0 5000 10000Position

0.0

0.1

0.2

0.3

0.4

0.5

Density

(b)

Figure 18: Mean density of spiders in each of the together (blue circles) and apart (red triangles)strides at tmax = 107, for r = 1 (a) and for r = 0.05 (b).

This can be considered an emergent phenomenon that arises as a means to increase spiderpacking close to the injection source. Spiders in this high density region rarely get an opportu-nity to spread their legs into the apart stride because both neighboring sites are almost alwaysoccupied. Also of note, at the distances where the linear decrease in spider density is no longerapparent, the distributions of strides becomes equal again. This equality of stride distributionindicates that the spiders are no longer experiencing the extreme exclusionary pressure observednear the injection site. Instead, the spiders on the periphery are able to act more like single spiders,which have an equal distribution in strides. These effects are again apparent at all time scales, asseen in Figure 19.

5.3.2 Density of leading spiders

Unlike the interior spiders, which never see a substrate, the leading spiders Ls and Rs are stronglyaffected by the enzymatic rate r. When spiders are in the boundary metastate (Sec. 2.2), theymove ballistically away from the origin, and the smaller the value of r, the less chance they haveof exiting the boundary state and returning to the diffusive D state. Figures 20 and 21 showthe probability distribution of µ(Rs) (the leading spiders location) for both r = 1 spiders andr = 0.05 spiders. Particularly at the shorter times in Figure 21, there is a distinct difference inthe distributions shape, with the r < 1 distributions having much longer tails, and distinctlynon-Gaussian shape. The mean of the distributions is the msd(t), reported in Sec. 4.1.1, whichgrows much faster for the r < 1 multi-spider systems than for r = 1. However, the completedistributions shown in Figures 20 and 21 reveal more information, particularly that the tails ofthe distribution are much shorter on the left than the right. This arises from the exclusionarypressures exerted by the next leading spider R′

s. However, the results of Sec. 5.1 show that despitethis exclusionary pressure, the distance between the two leading spiders 〈Neff(t)〉 continues toincrease as

√t regardless of the value of r. Hence, in the distributions at tmax in Fig. 20, there is

less distinction between the r = 1 and r < 1 walkers.

22

(a) (b)

Figure 19: Average density for spiders with legs together (blue circles) and apart (red triangles)plotted at several instants, for r = 1 (a) and for r = 0.05 (b).

0 2000 4000 6000 8000 10000Position

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

Pro

babili

ty d

ensi

ty

(a)

0 2000 4000 6000 8000 10000Position

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

Pro

babili

ty d

ensi

ty

(b)

Figure 20: Average density of the leading spider at tmax, for r = 1 (a) and for r = 0.05 (b).

23

(a) (b)

Figure 21: Average spider density of the leading spider plotted at several instants, for r = 1 (a)and for r = 0.05 (b).

6 Importance of Multi-Pedal Gaits for Transport

Many molecular walkers, including the natural motors kinesin and dynein, are multivalent—theyhave two (or more) attachment sites. Interestingly, it has been shown that in the AK spider model,the superdiffusive effects are only present when the number of legs k ≥ 2. Without the constraintsimposed by multiple legs the residence time-bias at the boundary when r < 1 does not lead to abias in motion towards substrates.

The multi-spider model has two potential sources of bias to cause superdiffusive motion of theleading spiders: the residence time bias at the boundary when r < 1, and the effective bias causedby the exclusionary pressure of the interior spiders. We know that the multi-spider systems aretransiently superdiffusive even when r = 1 due to the exclusionary pressure, but what happensto their collective behavior when they have only a single leg? In fact when k = 1 and r = 1 anAK spider is equivalent to an ordinary random walker that moves left and right with rate 1. Thus,we measured msd(t) for the leading walker of the multi-spider model with k = 1 and r = 1. Thevalues for msd(t) are compared with the multi-spiders model with k = 2 and r ∈ 1.0, 0.1, andshown on a log-log scale in Fig. 22, and the corresponding values of α(t) are shown in Fig. 23.The k = 1 walkers do exhibit transient superdiffusive behavior, but their values of αmax and td aresurpassed by k = 2 spiders when r < 0.1, as summarized in Table 3. Spiders with k = 1 achievemaximum msd(t) when r = 1 [1, 2], so by using r = 1 in our comparison we are comparing withthe most efficient single-legged spiders possible.

It is, however, necessary to make an adjustment of scales to correctly compare the msd(t)values between k = 2 and k = 1 walkers. Since the position of a spider is defined as the mean ofits attached leg positions, the k = 2 spiders move by only distance 0.5 with each step, in contrastto the k = 1 spiders which move by distance 1. Thus, in the analysis of k = 1 spiders shownin Figures 22 and 23, the k = 1 spiders move over a lattice with site spacing 0.5. In essence thiscorrection can be thought of as adjusting the diffusion constant of the k = 1, r = 1 walkers whichhave D = 1 to that of the k = 2, r = 1 walkers which have D = 0.5.

24

100 101 102 103 104 105 106 107

Time

10-1

100

101

102

103

104

105

106

107

108

109

msd

(t)

k=1, r=1.0k=2, r=1.0k=2, r=0.1

Figure 22: Comparison of msd(t) for the leading spider Rs in multi-spider simulations versus thek = 1 multi-spider model (with corrected diffusion constant of D = 0.5.)

100 101 102 103 104 105 106 107 108

Time

0.2

0.4

0.6

0.8

1.01.11.2

1.4

1.6

1.8

2.0

(t)

k=1, r=1.0k=2, r=1.0k=2, r=0.1

Figure 23: Comparison of finite difference approximation of α(t) for the leading spider Rs in multi-spider simulations versus the k = 1 multi-spider model (with corrected diffusion constant ofD = 0.5.) Horizontal lines show the threshold for ordinary diffusion at α = 1 and our definedthreshold for superdiffusion at α = 1.1.

25

0 500 1000 1500 2000 2500 3000N

0

50

100

150

200

250

300

Neff

r=1.0, data

r=1.0, Neff =0.29t0.81

r=0.5, data

r=0.5, Neff =0.26t0.86

r=0.1, data

r=0.1, Neff =0.27t0.87

r=0.05, data

r=0.05, Neff =0.21t0.90

r=0.01, data

r=0.01, Neff =0.03t1.11

r=0.005, data

r=0.005, Neff =0.01t1.32

Figure 24: Size of the effective product sea 〈Neff(N)〉 as a function of the number of visited sitesN.

7 Analysis of Maximum Product Sea Size

Simulation results presented in Sec. 4.1.1 suggest that the leading spider Rs moves diffusively inthe long time limit with the value of α(tmax) ≈ 1. Section 5.1 showed that this happens becausethe mean effective size of the product sea as seen by Rs, 〈Neff(t)〉, grows with time approximatelyas

√t. Hence, the duration of the D states 〈τD(t)〉 also grows with time, leading to asymptotically

diffusive motion. Furthermore the effective product sea size Neff can also be understood as being afunction N, the number of sites cleaved. Fig. 24 shows simulation estimates for 〈Neff(N)〉, which attimes close to tmax is almost linear. Thus, while the leading spider is cleaving sites at the boundary,the interior spiders are not following closely enough and the leading spider sees an increasinglylarge effective product sea. If a multi-spider system were to keep the leading spider superdiffusiveas t → ∞, it would have to ensure that 〈Neff(N)〉 does not grow too fast. The asymptotic boundthat ensures this property can be found analytically.

The expected exit time for a random walker from an interval (0, M) with two absorbing bound-aries at 0 and M is

〈Te(x)〉 = x(M − x)2

, (20)

where x is the starting position of the walker. As shown in Fig. 12 and explained in Sec. 5.1, theexpected exit time from an interval with one absorbing and one reflecting boundary is the sameas exit time from a interval with two absorbing boundaries of twice the size. Furthermore, whena spider moves over a region of product sites, its body position µ(F) moves like a simple randomwalker with a step size of 1/2.

When at time t the Rs spider moves off the boundary and into the D state, it enters a product

26

sea of expected size 〈Neff(N)〉 with one absorbing and one reflecting boundary. The expected timeto exit this interval is the expected duration of the D-state, τD. This can be found by using Eq. 20with M = 4〈Neff(N)〉 − 5 and x = 4〈Neff(N)〉 − 8, which gives

〈τD(N)〉 = 〈Te〉 = 3(4〈Neff(N)〉 − 8)2

. (21)

From Ref. [1], the average time interval during which the number of visited sites grows fromN + 3 to N + 4 is

〈τN〉 = 1r+

1 + r2 + r

〈Te〉, (22)

and the expected time to visit N + 3 sites is

〈T(N)〉 =N−1

∑i=0

〈τN〉. (23)

Thus, we can write 〈Neff(N)〉 to be a function of N, and by substituting Eq. 21 into the sum inEq. 23, we obtain

〈T(N)〉 = Nr+

1 + r2 + r

N−1

∑i=0

3(4〈Neff(i)〉 − 8)2

. (24)

Equation 24 shows that if 〈Neff(N)〉 = Θ(N), as Fig. 24 suggests, then 〈T(N)〉 = Θ(

N2),which corresponds to diffusive motion. Hence, in order for the leading spider to be superdiffusiveas t → ∞, we require 〈T(N)〉 = o(N2), which implied 〈Neff(N)〉 = o(N). Unfortunately, asFig. 12 shows, this is not the case for the multi-spider model—to maintain superdiffusive motionasymptotically, we need a mechanism stronger than passive injection at the origin.

8 Discussion

Collective and cooperative behaviors are essential to the motion of many natural molecular motorsand cellular transport systems, yet the incredible complexity of natural motor systems makes itdifficult to discern what types of walker interactions are necessary or sufficient for useful transportbehavior. In this work we begin to address these fundamental questions of nanoscale cooperationby investigating a much simpler model of walker motion based on the kinetic and mechanicalproperties of DNA-enzyme-based molecular spiders. The AK model of spider motion, while lack-ing in chemical detail, is able to encompass those features of spider motion that are necessaryfor superdiffusive motion of single spiders in 1D (i.e., irreversibility of substrate modification,multiplicity of legs, and a kinetic bias between modified and unmodified sites). These seem-ingly simple properties have previously been shown to lead to useful superdiffusive motion [15].Following this bottom-up approach to the investigation of spider motion, we have analyzed thecollective behavior of multiple identical spiders as they cooperate using only simple exclusionaryinteractions. Clearly, more complex interactions are possible, but we have shown that this singleadditional feature allows spider systems to move cargo in 1D superdiffusively over significantlylonger times and distances than can be accomplished with a single spider system (Sec. 4.1.1). Mul-tiple spiders with small values of r achieve higher values of αmax and td than either single AKspiders (Sec. 4.2) or cooperative single-legged spiders (Sec. 6). However, in the asymptotic limit as

27

t → ∞, we find that α(t) → 1, and the leading spiders of the multi-spider model move diffusivelyfor all values of r.

Analysis of the AK model shows that this asymptotic decay to diffusion is also a fundamentalproperty of the single spiders, and previous work has shown that the diffusive behavior resultsfrom the switching of single AK spiders between two metastates that partition the process intoalternating diffusive (D) and boundary (B) periods. Spiders move diffusively over products inthe D metastate, and ballistically away from the origin while they are attached to energy-richsubstrates in the B state. However, the duration of the B periods 〈τB(t)〉 is constant in time, whilethe duration of D periods 〈τD(t)〉 = Θ

(√t)

grows with time because the sea of products thatthe spider cleaves out also grows with time, and it takes increasingly long for spiders to exit thisenergy-devoid product sea.

A simple idea motivated the multi-spider model: if the effective size of the product sea Neffcould be sufficiently limited, then it would prevent the 〈τD(t)〉 from growing with time and hencelet the spider move asymptotically diffusively. This ends up not being the case, and even withnew spiders injected at the origin as fast as possible, without violating the exclusionary propertiesof spiders. We still found 〈Neff(t)〉 = O

(√N)

for all values of r (Sec. 5.1). The reasons behind thiscan be understood from several perspectives presented in this work.

One way to understand this result is to note the number of spiders released with time was〈S(t)〉 = O

(√t)

and largely independent of r, even with the injection rate λ = ∞ (Sec. 5.2). Underasymptotically superdiffusive motion of the leading spiders we would see the number of productscleaved N(t) = ω(

√t), and to fill this product sea would require S(t) = Θ(N(t)) = ω(

√t), which

is not achieved by the multi-spider model. Thus, we cannot seem to release spiders fast enoughto support superdiffusion indefinitely. This failure can be partly understood by observing that theonly spiders that actually get to attach to and cleave the energy bearing substrates are the leadingRs and Ls spiders. The other, interior spiders only ever walk on products. Thus, while thereis some bias exerted on interior spiders by the exclusionary pressure of injected spiders near theorigin, for the most part the motion of interior spiders is governed by diffusion. Hence, they do notmove fast enough to get clear of the injection site at the origin to allow enough other spiders to beinjected fast enough. Indeed, the density of spiders (Sec. 5.3) around the origin is nearly maximal,and also seemingly independent of r. Thus, it does not really matter how small the value of r is(and hence how much biasing energy is contained within substrates), because the interior spiderssee none of that energy and their diffusive motion is hence independent of r. Yet, the motion ofthese interior spiders remains the limiting factor for the injection rate of the new spiders neededto assist the leading spiders by reducing the effective size of the product sea. Hence, no matterhow fast the leading spiders are able to move initially, they will inevitably be hindered by theinsufficiently fast dispersal of the energy-deprived interior spiders.

Additionally, the failure of the multi-spider system to limit the effective product sea size canbe seen with the analytical approach of Sec. 7. This shows us that a spider system that can ensurethat 〈Neff(N)〉 = o(N) will allow asymptotically faster-than-diffusion transport. Future work willconcentrate on understanding how spider systems can achieve these bounds on the effective prod-uct sea size, potentially by supplying energy to the system from an external source. After all, thelack of energy available to the interior spiders seems to be the limiting factor for the cooperativetransport behaviors of the multi-spider model.

Despite these asymptotic results, it should be remembered that all practical nanoscale trans-

28

port problems for which molecular walkers are applicable take place over finite times and finitedistances and both single and multi-spider systems show potential for faster-than-diffusion trans-port for such applications. Other interesting applications of spiders which we are currently in-vestigating include cooperative 2D behavior for transport, and other chemically plausible modesof interaction between spiders and surfaces that are more complex than the simple exclusionaryprocesses examined in this work.

Acknowledgments. We thank Paul L. Krapivsky for insights into the behavior of multiple ran-dom walkers with an infinitely strong source [9]. This material is based upon work supported bythe National Science Foundation under grants 0829896 and 1028238.

References

[1] Antal, T., Krapivsky, P.L.: Molecular spiders with memory. Physical Review E 76(2), 021121(2007)

[2] Antal, T., Krapivsky, P.L., Mallick, K.: Molecular spiders in one dimension. Journal of Statis-tical Mechanics: Theory and Experiment 2007(08), P08027 (2007)

[3] Badoual, M., Julicher, F., Prost, J.: Bidirectional cooperative motion of molecular motors.PNAS 99(10), 6696–6701 (2002)

[4] Bortz, A.B., Kalos, M.H., Lebowitz, J.L.: A new algorithm for Monte Carlo simulation of Isingspin systems. Journal of Computational Physics 17(1), 10–18 (1975)

[5] Campas, O., Kafri, Y., Zeldovich, K.B., Casademunt, J., Joanny, J.F.: Collective dynamics ofinteracting molecular motors. Physical Review Letters 97, 038101 (2006)

[6] Frey, E., Parmeggiani, A., Franosch, T.: Collective phenomena in intracellular processes.Genome Informatics 15(1), 46–55 (2004)

[7] Hirokawa, N., Takemura, R.: Molecular motors and mechanisms of directional transport inneurons. Nature Reviews: Neuroscience 6(3), 201–214 (2005)

[8] Julicher, F., Ajdari, A., Prost, J.: Modeling molecular motors. Reviews of Modern Physics69(4), 1269–1281 (1997)

[9] Krapivsky, P.L.: Symmetric Exclusion Process with a Localized Source. arXiv:1208.3250[cond-mat.stat-mech] (2012)

[10] Lacasta, A.M., Sancho, J.M., Romero, A.H., Sokolov, I.M., Lindenberg, K.: From subdiffusionto superdiffusion of particles on solid surfaces. Phys. Rev. E 70(5), 051104 (Nov 2004)

[11] Lund, K., Manzo, A.J., Dabby, N., Michelotti, N., Johnson-Buck, A., Nangreave, J., Taylor,S., Pei, R., Stojanovic, M.N., Walter, N.G., Winfree, E., Yan, H.: Molecular robots guided byprescriptive landscapes. Nature 465, 206–210 (May 2010)

29

[12] Pei, R., Taylor, S.K., Stefanovic, D., Rudchenko, S., Mitchell, T.E., Stojanovic, M.N.: Behaviorof polycatalytic assemblies in a substrate-displaying matrix. Journal of the American Chemi-cal Society 128(39), 12693–12699 (2006)

[13] Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical recipes in C++. Cam-bridge University Press, New York, NY (2002)

[14] Schliwa, M., Woehlke, G.: Molecular motors. Nature 422(6933), 759–765 (2003)

[15] Semenov, O., Olah, M.J., Stefanovic, D.: Mechanism of diffusive transport in molecular spidermodels. Phys. Rev. E 83(2), 021117 (Feb 2011)

[16] Semenov, O., Olah, M.J., Stefanovic, D.: Multiple molecular spiders with a single localizedsource—the one-dimensional case. In: DNA 17: Proceedings of The Seventeenth Interna-tional Meeting on DNA Computing and Molecular Programming. Lecture Notes in Com-puter Science, vol. 6397, pp. 204–216. Springer (2011)

30